If you have Hawk as a contact you can convince Dr. Thomas to start supplying the Third Eye directly, but only if you have spread Eclipse around enough. Otherwise she tells you:

No, no, I don't think so. Taking over that contract would draw too much attention to ourselves. We need to prove we're indispensable before that might work. Help expand the network, though, and that'd be a major coup.

Seems like giving people eclipse expands the network faster than selling it them does - giving it to 5 people may succeed where selling it to 5 doesn't. More testing needed, perhaps. See discussion page.

For the purposes of Joining the third eye, you must distribute eclipse to at least 6 (or maybe 5 in some cases?) people. Distributing to more people afterwards has no known use, but you can gain some minor rewards in credits or items. The following are available for the purposes of this quest: